The Sylow subgroups of the absolute Galois group Gal(Q)

TL;DR

This paper characterizes the Sylow p-subgroups of the absolute Galois group of rationals, showing their structure as semidirect products involving free pro-p groups and p-adic integers, and solving related embedding problems.

Contribution

It provides a detailed description of the Sylow p-subgroups of Gal(Q) as semidirect products and proves the solvability of split embedding problems for these groups.

Findings

Sylow p-subgroups are semidirect products of Z_p acting on free pro-p groups.

Finite Z_p-quotients of free pro-p groups are classified.

Every split embedding problem for these groups is solvable.

Abstract

We describe the Sylow subgroups of Gal(Q) for an odd prime p, by observing and studying their decomposition as a semidirect product of Z_p acting on F, where F is a free pro-p group, and Z_p are the p-adic integers. We determine the finite Z_p-quotients of F and more generally show that every split embedding problem of Z_p-groups for F is solvable. Moreover, we analyze the Z_p-action on generators of F.